Twisted conjugacy classes of automorphisms of Baumslag-Solitar groups

Abstract

Let $\varphi :G\to G$ be a group endomorphism where $G$ is a finitely generated group of exponential growth, and denote by $R(\varphi )$ the number of twisted $\varphi$-conjugacy classes. Fel'shtyn and Hill \cite{fel-hill} conjectured that if $\varphi$ is injective, then $R(\varphi )$ is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups, see \cite{ll} and \cite {fel:1}. It was showed in \cite {gw:2} that the conjecture does not hold in general. Nevertheless in this paper, we show that the conjecture holds for the Baumslag-Solitar groups $B(m,n)$, where either $|m|$ or $|n|$ is greater than 1 and $|m|\ne |n|$. We also show that in the cases where $|m|=|n|>1$ or $mn=-1$ the conjecture is true for automorphisms. In addition, we derive few results about the coincidence Reidemeister number.

arXiv comment20 pages, new sections 6 and 7 are added

References in corpus (1)

• On the automorphism group of generalized Baumslag-Solitar groups

Cited by in corpus (8)

• On the automorphism group of generalized Baumslag-Solitar groups
• The lower central and derived series of the braid groups of the sphere and the punctured sphere
• A note on twisted conjugacy and generalized Baumslag-Solitar groups
• The $R_{\infty }$-property for right-angled Artin groups
• The geometry of twisted conjugacy classes in wreath products
• Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups
• Twisted conjugacy and quasi-isometry invariance for generalized solvable Baumslag-Solitar groups
• The Reidemeister spectrum of split metacyclic groups